## Geometric ergodicity in a weighted sobolev space

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##### Publication Date

2020-01##### Journal Title

Annals of Probability

##### ISSN

0091-1798

##### Publisher

Institute of Mathematical Statistics

##### Volume

48

##### Issue

1

##### Pages

380-403

##### Language

English

##### Type

Article

##### This Version

AM

##### Metadata

Show full item record##### Citation

Devraj, A., Kontoyiannis, I., & Meyn, S. (2020). Geometric ergodicity in a weighted sobolev space. Annals of Probability, 48 (1), 380-403. https://doi.org/10.1214/19-AOP1364

##### Abstract

For a discrete-time Markov chain $\{X(t)\}$ evolving on $\Re^\ell$ with
transition kernel $P$, natural, general conditions are developed under which
the following are established:
1. The transition kernel $P$ has a purely discrete spectrum, when viewed as a
linear operator on a weighted Sobolev space $L_\infty^{v,1}$ of functions with
norm, $$ \|f\|_{v,1} = \sup_{x \in \Re^\ell} \frac{1}{v(x)} \max \{|f(x)|,
|\partial_1 f(x)|,\ldots,|\partial_\ell f(x)|\}, $$ where $v\colon \Re^\ell \to
[1,\infty)$ is a Lyapunov function and $\partial_i:=\partial/\partial x_i$.
2. The Markov chain is geometrically ergodic in $L_\infty^{v,1}$: There is a
unique invariant probability measure $\pi$ and constants $B<\infty$ and
$\delta>0$ such that, for each $f\in L_\infty^{v,1}$, any initial condition
$X(0)=x$, and all $t\geq 0$: $$\Big| \text{E}_x[f(X(t))] - \pi(f)\Big| \le
Be^{-\delta t}v(x),\quad \|\nabla \text{E}_x[f(X(t))] \|_2 \le Be^{-\delta t}
v(x), $$ where $\pi(f)=\int fd\pi$.
3. For any function $f\in L_\infty^{v,1}$ there is a function $h\in
L_\infty^{v,1}$ solving Poisson's equation: \[ h-Ph = f-\pi(f). \] Part of the
analysis is based on an operator-theoretic treatment of the sensitivity process
that appears in the theory of Lyapunov exponents.

##### Identifiers

External DOI: https://doi.org/10.1214/19-AOP1364

This record's URL: https://www.repository.cam.ac.uk/handle/1810/295116

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